Teorema do ponto fixo de Lefschetz

Teorema do ponto fixo de Lefschetz (Redirected from Lefschetz–Hopf theorem) Jump to navigation Jump to search This article includes a list of general references, mas faltam citações em linha correspondentes suficientes. Ajude a melhorar este artigo introduzindo citações mais precisas. (Marchar 2022) (Saiba como e quando remover esta mensagem de modelo) Na matemática, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space {estilo de exibição X} to itself by means of traces of the induced mappings on the homology groups of {estilo de exibição X} . It is named after Solomon Lefschetz, who first stated it in 1926.

The counting is subject to an imputed multiplicity at a fixed point called the fixed-point index. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).

Conteúdo 1 Declaração formal 2 Esboço de uma prova 3 Lefschetz–Hopf theorem 4 Relation to the Euler characteristic 5 Relation to the Brouwer fixed-point theorem 6 Historical context 7 Frobenius 8 Veja também 9 Notas 10 Referências 11 External links Formal statement For a formal statement of the theorem, deixar {displaystyle fcolon Xrightarrow X,} be a continuous map from a compact triangulable space {estilo de exibição X} to itself. Define the Lefschetz number {estilo de exibição Lambda _{f}} do {estilo de exibição f} por {estilo de exibição Lambda _{f}:=soma _{kgeq 0}(-1)^{k}matemática {tr} (f_{*}|H_{k}(X,mathbb {Q} )),} the alternating (finito) sum of the matrix traces of the linear maps induced by {estilo de exibição f} sobre {estilo de exibição H_{k}(X,mathbb {Q} )} , the singular homology groups of {estilo de exibição X} with rational coefficients.

A simple version of the Lefschetz fixed-point theorem states: E se {estilo de exibição Lambda _{f}neq 0,} então {estilo de exibição f} has at least one fixed point, ou seja, there exists at least one {estilo de exibição x} dentro {estilo de exibição X} de tal modo que {estilo de exibição f(x)=x} . Na verdade, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to {estilo de exibição f} has a fixed point as well.

Note however that the converse is not true in general: {estilo de exibição Lambda _{f}} may be zero even if {estilo de exibição f} has fixed points.

Sketch of a proof First, by applying the simplicial approximation theorem, one shows that if {estilo de exibição f} has no fixed points, então (possibly after subdividing {estilo de exibição X} ) {estilo de exibição f} is homotopic to a fixed-point-free simplicial map (ou seja, it sends each simplex to a different simplex). This means that the diagonal values of the matrices of the linear maps induced on the simplicial chain complex of {estilo de exibição X} must be all be zero. Then one notes that, no geral, the Lefschetz number can also be computed using the alternating sum of the matrix traces of the aforementioned linear maps (this is true for almost exactly the same reason that the Euler characteristic has a definition in terms of homology groups; see below for the relation to the Euler characteristic). In the particular case of a fixed-point-free simplicial map, all of the diagonal values are zero, and thus the traces are all zero.

Lefschetz–Hopf theorem A stronger form of the theorem, also known as the Lefschetz–Hopf theorem, afirma que, E se {estilo de exibição f} has only finitely many fixed points, então {soma de estilo de exibição _{xin mathrm {Fixar} (f)}eu(f,x)=Lambda _{f},} Onde {matemática de estilo de exibição {Fixar} (f)} is the set of fixed points of {estilo de exibição f} , e {estilo de exibição eu(f,x)} denotes the index of the fixed point {estilo de exibição x} .[1] From this theorem one deduces the Poincaré–Hopf theorem for vector fields.

Relation to the Euler characteristic The Lefschetz number of the identity map on a finite CW complex can be easily computed by realizing that each {estilo de exibição f_{ast }} can be thought of as an identity matrix, and so each trace term is simply the dimension of the appropriate homology group. Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space, which in turn is equal to the Euler characteristic {chi de estilo de exibição (X)} . Thus we have {estilo de exibição Lambda _{matemática {Eu iria} }=chi (X). } Relation to the Brouwer fixed-point theorem The Lefschetz fixed-point theorem generalizes the Brouwer fixed-point theorem, which states that every continuous map from the {estilo de exibição m} -dimensional closed unit disk {estilo de exibição D^{n}} para {estilo de exibição D^{n}} must have at least one fixed point.

Poderá ser visto da seguinte forma: {estilo de exibição D^{n}} is compact and triangulable, all its homology groups except {estilo de exibição H_{0}} are zero, and every continuous map {displaystyle fcolon D^{n}to D^{n}} induces the identity map {estilo de exibição f_{*}colon H_{0}(D^{n},mathbb {Q} )to H_{0}(D^{n},mathbb {Q} )} , whose trace is one; all this together implies that {estilo de exibição Lambda _{f}} is non-zero for any continuous map {displaystyle fcolon D^{n}to D^{n}} .

Historical context Lefschetz presented his fixed-point theorem in (Lefschetz 1926). Lefschetz's focus was not on fixed points of maps, but rather on what are now called coincidence points of maps.

Given two maps {estilo de exibição f} e {estilo de exibição g} from an orientable manifold {estilo de exibição X} to an orientable manifold {estilo de exibição Y} of the same dimension, the Lefschetz coincidence number of {estilo de exibição f} e {estilo de exibição g} is defined as {estilo de exibição Lambda _{f,g}=sum (-1)^{k}matemática {tr} (D_{X}circ g^{*}circ D_{S}^{-1}circ f_{*}),} Onde {estilo de exibição f_{*}} is as above, {estilo de exibição g_{*}} is the homomorphism induced by {estilo de exibição g} on the cohomology groups with rational coefficients, e {displaystyle D_{X}} e {displaystyle D_{S}} are the Poincaré duality isomorphisms for {estilo de exibição X} e {estilo de exibição Y} , respectivamente.

Lefschetz proved that if the coincidence number is nonzero, então {estilo de exibição f} e {estilo de exibição g} have a coincidence point. He noted in his paper that letting {displaystyle X=Y} and letting {estilo de exibição g} be the identity map gives a simpler result, which we now know as the fixed-point theorem.

Frobenius Let {estilo de exibição X} be a variety defined over the finite field {estilo de exibição k} com {estilo de exibição q} elements and let {estilo de exibição {bar {X}}} be the base change of {estilo de exibição X} to the algebraic closure of {estilo de exibição k} . The Frobenius endomorphism of {estilo de exibição {bar {X}}} (often the geometric Frobenius, or just the Frobenius), denotado por {estilo de exibição F_{q}} , maps a point with coordinates {estilo de exibição x_{1},ldots ,x_{n}} to the point with coordinates {estilo de exibição x_{1}^{q},ldots ,x_{n}^{q}} . Thus the fixed points of {estilo de exibição F_{q}} are exactly the points of {estilo de exibição X} with coordinates in {estilo de exibição k} ; the set of such points is denoted by {estilo de exibição X(k)} . The Lefschetz trace formula holds in this context, and reads: {displaystyle #X(k)=soma _{eu}(-1)^{eu}matemática {tr} (F_{q}^{*}|H_{c}^{eu}({bar {X}},mathbb {Q} _{bem })).} This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, do {estilo de exibição {bar {X}}} with values in the field of {ell de estilo de exibição } -adic numbers, Onde {ell de estilo de exibição } is a prime coprime to {estilo de exibição q} .

Se {estilo de exibição X} is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius {displaystyle Phi _{q}} , which acts as the inverse of {estilo de exibição F_{q}} on cohomology: {displaystyle #X(k)=q^{dim X}soma _{eu}(-1)^{eu}matemática {tr} ((Phi _{q}^{-1})^{*}|H^{eu}({bar {X}},mathbb {Q} _{bem })).} This formula involves usual cohomology, rather than cohomology with compact supports.

The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields.

See also Fixed-point theorems Lefschetz zeta function Holomorphic Lefschetz fixed-point formula Notes ^ Dold, Albrecht (1980). Lectures on algebraic topology. Volume. 200 (2ª edição). Berlim, Nova york: Springer-Verlag. ISBN 978-3-540-10369-1. SENHOR 0606196., Proposition VII.6.6. References Lefschetz, Solomon (1926). "Intersections and transformations of complexes and manifolds". Transações da American Mathematical Society. 28 (1): 1–49. doi:10.2307/1989171. JSTOR 1989171. SENHOR 1501331. Lefschetz, Solomon (1937). "On the fixed point formula". Anais da Matemática. 38 (4): 819–822. doi:10.2307/1968838. JSTOR 1968838. SENHOR 1503373. links externos "Lefschetz formula", Enciclopédia de Matemática, Imprensa EMS, 2001 [1994] Controle de autoridade: National libraries Germany Categories: Fixed-point theoremsContinuous mappingsTheorems in algebraic topology